A three-dimensional generalization of QRT maps

TL;DR

This paper introduces a geometric method to construct three-dimensional birational maps that extend QRT maps, preserving certain quadrics, and demonstrates their application to discretizations of classical Nambu systems.

Contribution

It presents a novel geometric construction of 3D birational maps generalizing QRT maps, with conditions for degree reduction and applications to discretizations of Nambu systems.

Findings

Maps act as compositions of involutions along quadrics

Conditions for degree reduction to 3 are established

Applications include discretizations of Euler top and Zhukovski-Volterra gyrostat

Abstract

We propose a geometric construction of three-dimensional birational maps that preserve two pencils of quadrics. The maps act as compositions of involutions, which, in turn, act along the straight line generators of the quadrics of the first pencil and are defined by the intersections with quadrics of the second pencil. On each quadric of the first pencil, the maps act as two-dimensional QRT maps. While these maps are of a pretty high degree in general, we find geometric conditions which guarantee that the degree is reduced to 3. The resulting degree 3 maps are illustrated by two known and two novel Kahan-type discretizations of three-dimensional Nambu systems, including the Euler top and the Zhukovski-Volterra gyrostat with two non-vanishing components of the gyrostatic momentum.